Timeline for Illuminating piecewise-flat manifolds with geodesics
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
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| Feb 3, 2022 at 13:05 | history | edited | Martin Sleziak | CC BY-SA 4.0 |
a minor typo
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| Nov 7, 2015 at 22:38 | comment | added | Matheus | @JosephO'Rourke: The genus 3 flat surface described by Dmitri is known as 'Eierlegende Wollmilchsau'. You can find concrete models in Figure 1 of these papers here (arxiv.org/pdf/math/0509195.pdf) and here (arxiv.org/pdf/0912.1425.pdf). Also, this short Oberwolfach report paper of Thierry Monteil here (web.archive.org/web/20140422060503/http://www2.lirmm.fr/…) is an interesting reading. | |
| Aug 5, 2010 at 12:47 | comment | added | Joseph O'Rourke | @Per: Thanks, that helps considerably! | |
| Aug 5, 2010 at 5:29 | comment | added | Per Vognsen | Continued: The branch cuts are the transition loci for up and down ramps between the sheets. Each branch cut bumps the genus up by one. If you also identify the square's edges in the customary fashion, you get another bump, for a total genus of three. Now there's no edges left, so it's a manifold without boundary. | |
| Aug 5, 2010 at 5:20 | comment | added | Per Vognsen | Joseph: It might help to first look at the construction in terms of Dmitri's original description. Take two sheets of a unit square. In each of them join the branch points by horizontal slits (branch cuts). Imagine starting at the lower left corner of one sheet and circling the nearest branch point counterclockwise until you hit the branch cut. At that point you go up to the next sheet. If you continue the circling and now hit the branch cut one more time, you go back to the lower sheet. There's a lifting principle that shows that geodesics on the cover must project to geodesics on the base. | |
| Aug 4, 2010 at 23:29 | comment | added | Joseph O'Rourke | I am accepting this as the answer even though I have to admit I do not entirely understand how the 8 squares are identified along their 32 edges. But this is my fault, not Dmitri's. I need to study translation surfaces! | |
| Aug 4, 2010 at 23:26 | vote | accept | Joseph O'Rourke | ||
| Jul 22, 2010 at 0:55 | comment | added | Victor Protsak | Dmitri: Beautiful construction! Joseph: At the branch points the total angle is $2\times 2\pi=4\pi,$ so geodesic is forbidden to pass through them according to your rules. | |
| Jul 21, 2010 at 22:54 | comment | added | Joseph O'Rourke | @Dmitri: Thanks for the explanation. I will ponder this! | |
| Jul 21, 2010 at 22:52 | history | edited | Dmitri Panov | CC BY-SA 2.5 |
added 251 characters in body
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| Jul 21, 2010 at 22:49 | comment | added | Dmitri Panov | @Joseph, this is terminology from the theory of Riemman surfaces, en.wikipedia.org/wiki/Ramified_covering_map . Double ramified cover is a ramified cover of degree two, i.e. generic point has two preimages. An example of such a cover is $z\to z^2$, $z\in \mathbb C^1$. The example that I proposed can be understood without this treminology. You can glue $S$ from 8 squares of the size $1/2 \times 1/2$, at each vertex on $S$ $8$ squares should meet. | |
| Jul 21, 2010 at 22:35 | comment | added | Joseph O'Rourke | @Dmitri: Cool, I want to understand this, but the term "double ramified cover" is new to me and I am not finding a definition... | |
| Jul 21, 2010 at 21:43 | history | answered | Dmitri Panov | CC BY-SA 2.5 |